# Inner products on a Hilbert space

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## Main Question or Discussion Point

Hello,

I am taking a quantum mechanics course using the Griffiths textbook and encountering some confusion on the definition of inner products on eigenfunctions of hermitian operators. In chapter 3 the definition of inner products is explained as follows: $$\langle f(x)| g(x) \rangle = \int f(x)^{*} g(x)\, dx$$ Should you need to express some function $f(x)$ as a linear combination of functions $f_n(x)$ then the appropriate constants $c_n(x)$ can be found using Fouriers trick: $$c_n(x) = \langle f_n(x)| f(x) \rangle$$ This I understand. In chapter 4 this idea is applied to find the probability of measuring a certain spin of spin 1/2 particle in the x direction. The spinor $\chi$ will need to be expressed in the eigenfunctions of $\textbf{Sx}$: $\chi_{x+}$ and $\chi_{x-}$. So to find the appropriate coefficients one can apply fouriers trick again. $$c_+ = \langle \chi_{x+}| \chi\rangle$$ However when this inner product is calculated according to Griffiths: $$c_+ = \langle \chi_{x+}| \chi\rangle = (\chi_{x+})^{\dagger} \chi$$ It seems like the integral from the original definition has disappeared? Why is this? I understand you are working with vectors here instead of scalar valued functions, but does that change the defintion of the inner product on the Hilbert space? This result looks a lot more like the standard definition of the inner product, yet the first vector is also conjugated. Can someone explain this difference to me? Thanks!

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